string_theory

Then, the partition function which we build from the generalized action, (13.2), is
related to the partition function which we build the “flat” action, (13.1), by

Z = DX Dh e−(Sσ,F +V ),

= DXDh e−Sσ,F e−V ,

= DX Dh e−Sσ,F 1−V + 1 V 2 +··· , (13.4)
2

where Sσ,F is the flat action given in (13.1) and V is given by

V = 1 d2σ√−h hαβ∂αXµ∂βXνhµν (X). (13.5)
4πα

Now, the expression for V is called the vertex operator associated to the graviton state
of the string. Thus, we know that inserting a single copy of V in the path integral
corresponds to the introduction of a single graviton state. Inserting e−V into the path
integral corresponds to a coherent state of gravitons, changing the metric from ηµν to
ηµν + hµν(X). In this way we see that the background arbitrary metric in (13.2) is
indeed built of the quantized gravitons which arose from quantizing the closed string.

13.1.1 Conformal Invariance of Sσ and the Einstein Equations

We have seen that, in conformal gauge, the Polyakov action in a flat spacetime reduces
to a free theory. However, in a curved spacetime this is no longer true. In conformal
gauge, the worldsheet theory is described by an interacting two-dimensional field theory,

S = 1 d2σgµν (X )∂α X µ∂ α X ν . (13.6)
4πα

To understand these interactions in more detail, lets expand around a classical
solution which we take to simply be a string sitting at a point xµ,

√ (13.7)
Xµ(σ) = xµ + α Y µ(σ).

Here Y µ(σ) are the dynami√cal fluctuations around the point xµ, which we assume to
be small, and the factor of α is there for dimensional reasons. Now, expanding the
Lagrangian gives

gµν(X)∂αXµ∂αXν = α gµν (x) + √ gµν,ω (x)Y ω + α gµν,ωρ(x)Y ωY ρ + · · · ∂αY µ∂αY ν,
α 2

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where gµν,i1···in(x) ≡ ∂in · · · ∂i1 gµν (x). Note that each of the gµν,···’s appearing in the
Taylor expansion above are coupling constants for the interactions of the Y ’s. Also,
there are an infinite number of these coupling constants which are contained in gµν(X).

Now, classical, the theory defined by (13.6) is conformally invariant. However,
this is not true when we quantize the theory. To regulate divergences we will have to
introduce a UV cut-off and, typically, after renormalization, physical quantities depend
on the scale of a given process ξ. If this is the case, the theory is no longer conformally
invariant. There are plenty of theories which classically possess scale invariance which
is broken quantum mechanically. The most famous of these is Yang-Mills.

As has been shown throughout, conformal invariance in string theory is a very good
property, it is in fact a gauge symmetry. Thus, we need to see under what conditions
our theory, defined by (13.6), remains conformally invariant after quantization. These
conditions will be given by the coupling constants and whether they depend on ξ or not,
i.e. if the coupling constants do not depend on ξ then our theory will be conformally
invariant under the quantization process. And so, we need to see how the coupling
constants depend on ξ.

The object which describes how a coupling constant depend on a scale ξ is called
the β-function. Since we have a functions worth of couplings, we should really be
talking about a β-functional, schematically of the form

βµν (g) = ∂gµν (X; ξ ) . (13.8)
∂ ln(ξ)

Now, there is an easy way to express whether the quantum theory will be invariant or
not. It is defined by

βµν (g) = 0, (13.9)

i.e. if the β-function(al) vanishes then the quantum version of (13.6) will remain
conformally invariant. The only thing left to do is to see what restriction this constraint
of having the β-function(al) vanish imposes on the coupling constants, gµν, and, as we
will see, this restriction is more than incredible.

The strategy is as follows. We will isolate the UV-divergence of (13.6) and then
use this to see what kind of term we should add. The β-function will then vanish if
this term vanishes. So, let us now proceed.

To begin, note that around any point x, we can√always pick Riemann normal
coordinates such that the expansion in Xµ(σ) = xµ + α Y µ(σ) gives

gµν (X) = δµν − α Rµλνκ(x)Y λY κ + O Y αY γY δ . (13.10)
3

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Plugging this back into the action, (13.6), gives (up to quadratic order in the fluctua-

tions Y )

S = 1 d2σ ∂Y µ∂Y ν δµν − α RµλνκY λY κ∂Y µ∂Y ν . (13.11)
4π 3

Now, we can now treat this as an interacting quantum field theory in two dimensions.
The quartic interaction gives a vertex with the Feynman rule,

∼ Rµλνκ(kµ · kν), (13.12)

where kαµ is the two-momentum (α = 1, 2 is a worldsheet index) for the scalar field Y µ.
Now that we have reduced the problem to a simple interacting quantum field theory,

we can compute the β-function using whatever method we like. The divergence in the
theory comes from the one-loop diagram

. (13.13)

Let us now think of this diagram in position space. The propagator for a scalar particle

is given by

Y λ(σ)Y κ(σ ) = − 1 δλκ ln |σ − σ |2 . (13.14)
2

For the scalar field running in the loop, the beginning and end point coincide. The

propagator diverges as σ → σ , which is simply reflecting the UV divergence that we

would see in the momentum integral around the loop. To isolate this divergence, we

choose to work with dimensional regularization, with d = 2 + . The propagator then

becomes,

Y λ(σ)Y κ(σ ) = 2πδλκ d2+ k eik(σ−σ )
(2π)2+ k2
, (13.15)

and so we see that lim Y λ(σ)Y κ(σ ) −→ δλκ .

σ→σ (13.16)

The necessary counterterm for this divergence can be determined simply by replacing
Y λY κ in the action with Y λY κ . To subtract the 1/ term, we add the counterterm

given by

RµλνκY λY κ∂Y µ∂Y ν −→ RµλνκY λY κ∂Y µ∂Y ν − 1 Rµν ∂Y µ∂Y ν. (13.17)

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